Preheat oven to 350 F.
In a medium-sized mixing bowl or bowl of a stand mixer, mix together sugar, cake flour, baking soda, salt, and zest.
Add butter and mix on medium-low speed for three minutes. Because there is so little butter, you'll end up with a very fine crumb texture.
In a small mixing bowl, whisk together eggs, oil, and vanilla extract until smooth.
Add the egg mixture to the flour mixture and beat on medium speed until just combined.
Slowly add lemon juice and mix on low speed until just combined.
Fill cupcake liners 3/4 full.
Bake for 14 minutes and then test to see if they are done. They are done when a toothpick comes out without wet batter stuck to it. If they are not done, test again every two minutes.
When the cupcakes are done, remove them immediately from cookie sheet or cupcake tins and leave them on a cooling rack (or just on your counter if you don't own a cooling rack) to cool.
In a medium-sized mixing bowl, beat butter for 3 minutes on high speed until light and fluffy.
Slowly mix in powdered sugar until fully combined.
Mix in ginger and salt.
Spread or pipe on cooled cupcakes.